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## BASIC AC POWER## Power Triangle
In AC circuits, current and voltage are normally out of phase and, as a result, not all the power produced by the generator can be used to accomplish work. By the same token, power cannot be calculated in AC circuits in the same manner as in DC circuits. The power triangle, shown in Figure 1, equates AC power to DC power by showing the relationship between generator output (apparent power - S) in volt-amperes (VA), usable power (true power - P) in watts, and wasted or stored power (reactive power - Q) in volt-amperes-reactive (VAR). The phase angle (Θ) represents the inefficiency of the AC circuit and corresponds to the total reactive impedance (Z) to the current flow in the circuit. Figure 1 Power TriangleThe power triangle represents comparable values that can be used directly to find the efficiency level of generated power to usable power, which is expressed as the power factor (discussed later). Apparent power, reactive power, and true power can be calculated by using the DC equivalent (RMS value) of the AC voltage and current components along with the power factor. Menu
## Apparent Power
## True Power
## Reactive PowerReactive power (Q) is the power consumed in an AC circuit because of the expansion and collapse of magnetic (inductive) and electrostatic (capacitive) fields. Reactive power is expressed in volt-amperes-reactive (VAR). Equation (9-3) is a mathematical representation for reactive power. Unlike true power, reactive power is not useful power because it is stored in the circuit itself. This power is stored by inductors, because they expand and collapse their magnetic fields in an attempt to keep current constant, and by capacitors, because they charge and discharge in an attempt to keep voltage constant. Circuit inductance and capacitance consume and give back reactive power. Reactive power is a function of a systems amperage. The power delivered to the inductance is stored in the magnetic field when the field is expanding and returned to the source when the field collapses. The power delivered to the capacitance is stored in the electrostatic field when the capacitor is charging and returned to the source when the capacitor discharges. None of the power delivered to the circuit by the source is consumed. It is all returned to the source. The true power, which is the power consumed, is thus zero. We know that alternating current constantly changes; thus, the cycle of expansion and collapse of the magnetic and electrostatic fields constantly occurs. ## Total PowerThe ## Power FactorPower factor (pf) is the ratio between true power and apparent power. True power is the power consumed by an AC circuit, and reactive power is the power that is stored in an AC circuit. CosΘ is called the power factor (pf) of an AC circuit. It is the ratio of true power to apparent power, where Θ is the phase angle between the applied voltage and current sine waves and also between P and S on a power triangle (Figure1). Equation (9-4) is a mathematical representation of power factor. Power factor also determines what part of the apparent power is real power. It can vary from 1, when the phase angle is 0°, to 0,when the phase angle is 90°. In an inductive circuit, the current lags the voltage and is said to have a lagging power factor, as shown in Figure 2. Figure 2 Lagging Power FactorIn a capacitive circuit, the current leads the voltage and is said to have a leading power factor, as shown in Figure 3. Figure 3 Leading Power FactorA mnemonic memory device, "ELI the ICE man," can be used to remember the voltage/current relationship in AC circuits. ELI refers to an inductive circuit (L) where current (I) lags voltage (E). ICE refers to a capacitive circuit (C) where current (I) leads voltage (E). ## Power in Series R-L CircuitExample: A 200Ω resistor and a 50 Ω are placed in series with a voltage source, and the total current flow is 2 amps, as shown in Figure 4. Find: - pf
- applied voltage,V
- P
- Q
- S
Figure 4 Series R-L CircuitSolution: ## Power in Parallel R-L CircuitExample: A 600 Ω resistor and 200 Ω X Find: - IT
- pf
- P
- Q
- S
Figure 5 Parallel R-L CircuitSolution: ## Power in Series R-C CircuitExample: An 80 Ω X Find: - Z
- IT
- pf
- P
- Q
- S
Figure 6 Series R-C CircuitSolution: ## Power in Parallel R-C CircuitExample: A 30 Ω resistance and a 40 Ω X Find: - IT
- Z
- pf
- P
- Q
- S
Figure 7 Parallel R-C CircuitSolution: ## Power in Series R-C-L CircuitExample: An 8 Ω resistance, a 40 Ω X Find: - Z
- VT
- pf
- P
- Q
- S
Figure 8 Series R-C-L Circuit## Power in Parallel R-C-L CircuitsExample: An 800 Ω resistance, 100 Ω X Figure 9 Parallel R-C-L CircuitSolution: ## Three-Phase Circuits
## Three-Phase SystemsA three-phase (3φ) system is a combination of three single-phase systems. In a 3φ balanced system, power comes from a 3φ AC generator that produces three separate and equal voltages, each of which is 120° out of phase with the other voltages (Figure 10). Figure 10 Three-Phase ACThree-phase equipment (motors, transformers, etc.) weighs less than single-phase equipment of the same power rating. They have a wide range of voltages and can be used for single-phase loads. Three-phase equipment is smaller in size, weighs less, and is more efficient than single-phase equipment. Three-phase systems can be connected in two different ways. If the three common ends of each phase are connected at a common point and the other three ends are connected to a 3φ line, it is called a wye, or Y-, connection (Figure 11). If the three phases are connected in series to form a closed loop, it is called a delta, or Δ-, connection. Figure 11 3φ AC Power Connections## Power in Balanced 3φ LoadsBalanced loads, in a 3φ system, have identical impedance in each secondary winding (Figure 12). The impedance of each winding in a delta load is shown as Z Figure 12 3φ Balanced LoadsIn a balanced delta load, the line voltage (V In a balanced wye load, the line voltage (V Because the impedance of each phase of a balanced delta or wye load has equal current, phase power is one third of the total power. Equation (9-10) is the mathematical representation for phase power (P Total power (P As you can see, the total power formulas for delta- and wye-connected loads are identical. Total apparent power (S Figure 13 3φ Power TriangleA balanced three-phase load has the real, apparent, and reactive powers given by: Example 1: Each phase of a delta- connected 3φ AC generator supplies a full load current of 200 A at 440 volts with a 0.6 lagging power factor, as shown in Figure 14. Find: - V
_{L} - I
_{L} - P
_{T} - Q
_{T} - S
_{T}
Figure 14 Three-Phase Delta GeneratorSolution: Example 2: Each phase of a wye- connected 3φ AC generator supplies a 100 A current at a phase voltage of 240V and a power factor of 0.9 lagging, as shown in Figure 15. Find: - V
_{L} - P
_{T} - Q
_{T} - S
_{T}
Figure 15 Three-Phase Wye GeneratorSolution: ## Unbalanced 3 |